Steven Williamson developed a model that seems to imply that quantitative easing results in deflation. Nick Rowe was very critical.
As Nick and others have pointed out, Williamson showed the weakness of his framing. Some real world market phenomenon cannot be usefully described by carefully modeling a long run equilibrium.
Williamson's model has an element of truth. Suppose a central bank wanted to have a large balance sheet. And further, it wanted to keep nominal interest rates low. How could it manage that? All it would need to do is generate expectations of deflation. This would raise the real interest rate that could be earned on its liabilities for any given nominal interest rate. That would increase the demand to hold its liabilities. And so, the central bank could issue more liabilities and hold a larger portfolio of assets. Why would it want to do that? Perhaps it wants to funnel more credit into the housing market than private investors consider wise.
None of this is news to Market Monetarists. But it has a bit of a "through the looking glass" aspect, because Market Monetarists are always making these arguments in reverse. Sumner is constantly arguing that low expected inflation results in a high demand for base money, and it is either explicitly or implicitly considered a bad thing. In Boom and Bust Banking, Jeff Hummel's contribution is a long attack on what appears to be an effort by the Fed to expand its balance sheet so that it can allocate credit. The key complaint is the payment of interest on reserves.
So, Market Monetarists would say that trying to generate a large balance sheet for the central bank should be avoided. Paying interest on reserve balances tends to raise the demand for central bank reserves, which we recognize means that a higher quantity of base money would be needed for macroeconomic equilibrium. The inference that Market Monetarists would draw is that the interest rate on reserves should be low, zero, or negative. We don't see such a low interest rate as valuable in and of itself, and so do not favor generating deflation to make the low interest rate consistent with long run equilibrium. Quite the contrary, we sometimes predict and perhaps just hope that a recovery of spending on output would allow credit markets to clear at higher real and nominal interest rates, including any interest rate paid on reserves. If anything, the thrust of Market Monetarist arguments is that one reason to target a more rapid trend growth rate of nominal GDP is to generate slightly higher nominal interest rates and a smaller equilibrium balance sheet for the central bank.
Why have Market Monetarists supported quantitative easing? The reason is to raise spending on output--nominal GDP. If the current growth path of nominal GDP is counted as an equilibrium, then the point is to generate a disequilibrium--an excess supply of base money-- that will be cleared up through an increase in the demand for nominal base money balances by an increase in spending on output--a higher growth path of prices, a higher growth path of real output, or some of both.
Nick Rowe finally emphasized this point, after it was pointed to by Robert Murphy. Williamson has the central bank seeking to have a large balance sheet in real terms, which must be done by making base money more attractive to hold. Paying higher nominal interest rates on reserves, or given whatever interest rate is paid, generating expectations of lower inflation, increase the real demand to hold base money balances. To the degree that quantitative easing is supposed to generate a higher growth path of for the price level, the goal is not to raise the central bank's balance sheet in real terms. The higher nominal quantity of base money is supposed to be held in the long run because the higher growth path of prices reduces real balances back to equilibrium.
However, pretty much all the Market Monetarists have been more and more inclined to see the current level of base money as more than adequate. (And how is that different than excessive?) The problem is the target. Creating large amounts of base money that will be soon removed has little effect on spending on output. If the Fed wants more spending on output, it needs to target a higher growth path for nominal GDP. Lower unemployment on the condition that inflation doesn't rise above 2.5% in the medium run just doesn't make it.
Oddly enough, the target is implicit in Williamson's framing too. If the "target" were for the central bank to have a large "real" balance sheet and keep nominal interest rates low, then it must generate expectations of deflation. The problem would be with this odd target.
And, of course, there is Williamson's odd notion that this would somehow actually result in goods and services being exchanged at progressively lower prices.